Noncommutative Spectral Geometry: A Short Review

We review the noncommutative spectral geometry, a gravitational model that combines noncommutative geometry with the spectral action principle, in an attempt to unify General Relativity and the Standard Model of electroweak and strong interactions. Despite the phenomenological successes of the model, the discrepancy between the predicted Higgs mass and the current experimental data indicate that one may have to go beyond the simple model considered at first. We review the current status of the phenomenological consequences and their implications. Since this model lives by construction at high energy scales, namely at the Grand Unified Theories scale, it provides a natural framework to investigate early universe cosmology. We briefly review some of its cosmological consequences.Comment: 11 pages. Invited talk in the Sixth International Workshop DICE2012, Castello Pasquini/Castiglioncello (Tuscany), September 17-21, 201

Aspects of the Bosonic Spectral Action: Successes and Challenges

A short introduction on elements of noncommutative geometry, which offers a purely geometric interpretation of the Standard Model and implies a higher derivative gravitational theory, is presented. Physical consequences of almost commutative manifolds are briefly discussed and cosmological consequences of the gravitational sector, which is shown not to be plagued by linear instability, are highlighted. Successes and challenges are discussed. A novel spectral action proposal based on zeta function regularisation is briefly presented.Comment: 10 pages. Invited talk to appear in the Proceedings of the Corfu Summer Institute 2015 "School and Workshops on Elementary Particle Physics and Gravity", 1-27 September 2015, Corfu, Greec

The role of topologigal defects in cosmology

Topological defects are involved in a plethora of physical and astrophysical phenomena. In these lectures, I will review the r\^ ole they could play in the large-scale structure formation and the anisotropies of the cosmic microwave background, as well as in various high energy phenomena, including baryon number asymmetry, ultra-high energy cosmic rays, and gamma ray bursts. I will then summarize the gravitational effects of cosmic strings. Finally, I will briefly discuss the r\^ ole of topological defects in brane world cosmology.Comment: Invited lectures in the NATO ASI / COSLAB (ESF) School ``Patterns of symmetry breaking'', September 2002 (Cracow). 28 page

Inflation and cosmic (super)strings: implications of their intimate relation revisited

We briefly discuss constraints on supersymmetric hybrid inflation models and examine the consistency of brane inflation models. We then address the implications for inflationary scenarios resulting from the strong constraints on the cosmic (super)string tension imposed from the most recent cosmic microwave background temperature anisotropies data.Comment: 9 pages; Invited talk in the "Quantized Flux in Tightly Knotted and Linked Systems" workshop in association with the Newton Institute programme "Topological Dynamics in the Physical and Biological Sciences" -- Isaac Newton Institute for Mathematical Sciences, Cambridge 3 - 7 December 201

Aspects of the Bosonic Spectral Action

A brief description of the elements of noncommutative spectral geometry as an approach to unification is presented. The physical implications of the doubling of the algebra are discussed. Some high energy phenomenological as well as various cosmological consequences are presented. A constraint in one of the three free parameters, namely the one related to the coupling constants at unification, is obtained, and the possible role of scalar fields is highlighted. A novel spectral action approach based upon zeta function regularisation, in order to address some of the issues of the traditional bosonic spectral action based on a cutoff function and a cutoff scale, is discussed.Comment: 16 pages, Invited talk in the Fourth Symposium on Prospects in the Physics of Discrete Symmetries, DISCRETE 2014, King's College London,2-6 December 201